Question

Using MO theory, predict which of the following species has the longest bond and which the strongest bond, respectively: O2, O2−, O22−

Answer 1

MO theory stands for Molecular Orbital theory which is sued to detremine the molecule structure. The bond length in the oxygen species, O2+, O2, O2-, O2^(2-), can be explained by the positions of the electrons in this theory. 
bond order = 1/2 (#e- in bonding MO's - #e- in antibonding MO's) 
bond order for O2 is 2 
bond order for O2+ is 2.5
bond order for O2- is 1.5
bond order for O2(2-) is 1

Answer 2

MO theory i.e. molecular orbital theory is used to determine the structure of molecule and explains the bonding present in the molecule by using linear combination of atomic orbitals. The bond length in the given species of oxygen can be identify on the basis of the positions of the electrons in this theory.

The mathematical expression of bond order is given by:

$Bond order = \frac{1}{2}\times (electrons in bonding Molecular orbitals - electrons in anti-bonding Molecular orbitals)$

The molecular diagrams are shown in the image.

Now, calculate bond orders:

For, $O_{2}^{2-}$:  It consist of 10 bonding electrons and 8 anti-bonding electrons.

Bond order = $\frac{1}{2}\times (10 - 8)$

= 1

For, $O_{2}$:  It consist of 10 bonding electrons and 6 anti-bonding electrons.

Bond order = $\frac{1}{2}\times (10 - 6)$

= 2

For, $O_{2}^{-}$:  It consist of 10 bonding electrons and 7 anti-bonding electrons.

Bond order = $\frac{1}{2}\times (10 - 7)$

= 1.5

Since, bond order is inversely proportional to bond length implies higher the bond order short will be the bond length.

Thus, order of bond length: $O_{2}<  O_{2}^{-}<  O_{2}^{2-}$

Longest bond: $O_{2}^{2-}$